Bearing Capacity of Shallow Foundations Notes

Last reviewed 16 Sept 2026 · 7 min read

Definitions

DefinitionBearing capacity terms

Gross pressure intensity $q$ — total pressure at the base of the footing (structure load + footing + backfill).
Net pressure intensity $q_{n} = q - γ D_{f}$ — excess over the overburden removed.
Ultimate bearing capacity $q_{u}$ — minimum gross pressure that causes shear failure of the supporting soil.
Net ultimate bearing capacity $q_{n u} = q_{u} - γ D_{f}$ .
Net safe bearing capacity $q_{n s} = q_{n u} / F$ ( $F$ = factor of safety, usually 2.5 to 3).
Safe bearing capacity $q_{s} = q_{n s} + γ D_{f}$ .
Safe bearing pressure (net soil pressure for a permissible settlement) — pressure that causes settlement equal to the permissible value.
Allowable bearing pressure — the lesser of the net safe bearing capacity (shear criterion) and the safe bearing pressure (settlement criterion).

$D_{f}$ = depth of foundation below ground; a foundation is shallow when $D_{f} \leq B$ (roughly).

Modes of shear failure

Mode	Soil	Features
General shear failure	Dense sands, stiff clays (low compressibility)	Well-defined slip surfaces to the ground surface; bulging of ground; sudden failure with a clear peak on the load–settlement curve; tilting
Local shear failure	Medium dense sands, medium clays	Slip surfaces not reaching the surface; some bulging; no clear peak
Punching shear failure	Loose sands, soft clays, deep footings	Vertical shearing around the footing edge; large settlement; no heave

Terzaghi's bearing capacity theory

Assumptions: strip footing with a rough base; shallow ( $D_{f} \leq B$ ); soil above the base replaced by a surcharge $q = γ D_{f}$ (its shear strength ignored); homogeneous, isotropic soil; general shear failure; load vertical and concentric; Mohr–Coulomb strength.

The failure zone has three parts: an elastic wedge under the footing (angle $ϕ$ with the base), radial shear zones (log-spiral) and Rankine passive zones.

FormulaTerzaghi's equations (general shear)

Strip: $q_{u} = c N_{c} + q N_{q} + 0.5 γ B N_{γ}$

Square: $q_{u} = 1.3 c N_{c} + q N_{q} + 0.4 γ B N_{γ}$

Circular (diameter B): $q_{u} = 1.3 c N_{c} + q N_{q} + 0.3 γ B N_{γ}$

$N_{c}$ , $N_{q}$ , $N_{γ}$ = bearing capacity factors (functions of $ϕ$ ); $q = γ D_{f}$ .

Local shear failure: use $c^{'} = \frac{2}{3} c$ and $ϕ^{'} = tan^{- 1} (\frac{2}{3} tan ϕ)$ with the same equations.

$ϕ$	$N_{c}$ (Terzaghi)	$N_{q}$	$N_{γ}$
0°	5.7	1.0	0
10°	9.6	2.7	1.2
20°	17.7	7.4	5.0
30°	37.2	22.5	19.7
35°	57.8	41.4	42.4
40°	95.7	81.3	100.4

For $ϕ = 0$ (saturated clay, undrained): strip $q_{u} = 5.7 c + γ D_{f}$ .

IS 6403 general bearing capacity equation

Code ProvisionIS 6403 — net ultimate bearing capacity (general shear failure)

q_{n u} = c N_{c} s_{c} d_{c} i_{c} + q (N_{q} - 1) s_{q} d_{q} i_{q} + 0.5 γ B N_{γ} s_{γ} d_{γ} i_{γ} W^{'}

Shape factors (rectangle $B \times L$ ): $s_{c} = s_{q} = 1 + 0.2 \frac{B}{L}$ ; $s_{γ} = 1 - 0.4 \frac{B}{L}$ (square: 1.2, 1.2, 0.8; circle: 1.3, 1.2, 0.6).

Depth factors: $d_{c} = 1 + 0.2 \frac{D _{f}}{B} N_{ϕ}$ ; $d_{q} = d_{γ} = 1 + 0.1 \frac{D _{f}}{B} N_{ϕ}$ for $ϕ > 10°$ (= 1 for $ϕ < 10°$ ), with $N_{ϕ} = tan^{2} (45° + ϕ /2)$ .

Inclination factors (load inclined at $α$ to the vertical): $i_{c} = i_{q} = (1 - \frac{α}{90})^{2}$ ; $i_{γ} = (1 - \frac{α}{ϕ})^{2}$ .

Water table factor $W^{'}$ : 0.5 if the water table is at or above the base; 1.0 if it is at a depth $\geq B$ below the base; interpolate linearly between.

Bearing capacity factors (IS 6403)

$ϕ$	0°	5°	10°	15°	20°	25°	30°	35°	40°
$N_{c}$	5.14	6.49	8.35	10.98	14.83	20.72	30.14	46.12	75.31
$N_{q}$	1.00	1.57	2.47	3.94	6.40	10.66	18.40	33.30	64.20
$N_{γ}$	0.00	0.45	1.22	2.65	5.39	10.88	22.40	48.03	109.41

Effect of water table

A water table within the zone of shear reduces the effective unit weight: at or above the base, $γ$ in the third term becomes $γ^{'}$ (≈ half), i.e. $W^{'}$ = 0.5; above the base, the surcharge term also uses the effective stress at base level.

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Bearing Capacity of Shallow Foundations